Solutions of some quadratic Diophantine equations

✍ Scribed by Refik Keskin

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 249 KB
• Volume: 60
• Category: Article
• ISSN: 0898-1221
• DOI: 10.1016/j.camwa.2010.08.012

✦ Synopsis

In this study, we investigate positive integer solutions of the Diophantine equations x 2 -kxy∓y 2 ∓x = 0 and x 2 -kxy-y 2 ∓y = 0. It is shown that when k > 3, x 2 -kxy+y 2 +x = 0 has no positive integer solutions but the equation x 2 -kxy + y 2 -x = 0 has positive integer solutions. Moreover, it is shown that the equations x 2 -kxyy 2 ∓ x = 0 and x 2 -kxyy 2 ∓ y = 0 have positive solutions when k ≥ 1.